## 1. Introduction

A striking and ubiquitous feature of atmospheric flow is a meandering jet stream that is located at a break in the tropopause and almost girdles the globe. A related feature is the distinctive structure of the accompanying potential vorticity (PV) distribution. It takes the form of a band of enhanced PV gradient aligned along the jet and located on isentropic surfaces that transect the tropopause break. This band of enhanced PV gradient (∇_{θ}PV) constitutes the starting point of the present study.

_{θ}

^{2}

_{θ}

*U,*

*U*denotes the flow component directed along the jet axis. In effect the relationship indicates that a local maximum of ∇

_{θ}PV also connotes a maximum in the flow. Homogeneity of the ambient tropospheric PV field accords with the mixing of PV by transient eddies (e.g., McIntyre and Palmer 1984; Sun and Lindzen 1994), and likewise the enhanced gradient accords with PV frontogenesis at the tropopause break accompanying the evolution of these eddies (Davies and Rossa 1998). Note also that the amplitudes of the instantaneous and the time-mean ∇

_{θ}PV bands amount, respectively, to ∼50 × 10

^{−6}pvu m

^{−1}and ∼3 × 10

^{−6}PVU m

^{−1}(PVU: PV unit) and these values correspond to ∼

*O*(100) and ∼

*O*(10) enhancement of their ambient atmospheric or the purely

*β*-attributable gradient.

Irrespective of a band's origin its existence, structure, and amplitude carry important dynamical ramifications. First, Rossby waves propagate on the ∇_{θ}PV field, and the latter field's localized structure exerts a direct influence upon the dynamical properties and transmissivity of the waves. Second, the strength of the two types of bands imply that large-scale adiabatic displacement of air parcels across the band would produce PV anomalies of ∼7 PVU and ∼3 PVU, respectively, and in turn such anomalies would connote significant modifications of the in situ (and far field) flow and thermal distributions.

These inferences suggest that the band's structure, allied to its temporal persistence and streamwise length, marks it out as an important tropopause-level dynamical feature and a possible waveguide in a variety of flow settings. Two such settings are (i) the transient response to and interaction with a juxtaposed meso- or synoptic-scale vortexlike anomaly and (ii) the quasi-steady response to lower-level planetary-scale stationary forcing.

The motivation and rationale for considering the first setting arise from the ubiquity and amplitude of such anomalies. For example Fig. 3a, which shows the PV distribution on the 320-K surface for the same instant as Fig. 1, pinpoints the presence of meso- and synoptic-scale positive PV anomalies (features A, B, and C) and a negative anomaly (E) that are all in the vicinity of the ∇_{θ}PV band. [Some anomalies (D and F) are removed from the band]. In contrast, for the second setting the effect of the low-level orographic forcing would only be evident at higher elevations as an anticyclonic vorticity pattern, and there is a hint of such a signal in Fig. 3a over the northern part of the Rockies. A more vivid depiction of this orographic signal is evident in the time-mean vorticity field of Fig. 3b, which shows that the flow over the Rockies, Greenland, and the Himalayas is characterized by negative vorticity values that are almost unrivalled elsewhere in the extratropics.

The present heuristic study is designed to explore aspects of the ∇_{θ}PV band's influence in the foregoing two settings. The approach adopted is based upon first representing the band in the highly idealized form of a latitudinally and vertically aligned interface separating two regions of uniform (but different) potential vorticity. Then the response is evaluated when the interface is perturbed by an isolated vortex that is either fixed in space or advects with the ambient flow.

This idealized configuration has a long pedigree in atmospheric studies (Platzman 1949) and has been invoked in several recent studies (Polvani and Plumb 1992; Nakamura and Plumb 1994; Ambaum and Verkley 1995; Swanson et al. 1997; Pomroy and Thorpe 2000). It has the merit of restricting Eulerian in situ adiabatic PV changes to the neighborhood of the interface and the vortex and, moreover, in a simple two-dimensional setting it has been shown that the interface can sustain trapped Rossby waves. The configuration has also been adopted in oceanographic studies to model strong currents such as the Gulf Stream and such a current's meanders and interaction with an isolated vortex (see Bell 1990; Pratt et al. 1991 and references therein). In particular it has been shown (Bell 1990) that a point vortex can excite a resonant response of the trapped wave, and that a backreaction of perturbations on the interface induces a “latitudinal” displacement of the point vortex.

The paper is organized as follows. In section 2 the quintessential dynamics of a finite-scale vortex's influence upon the interface is examined in a simple linear *β*-plane setting. This setting is designed to complement, elaborate, and extend the results of earlier studies (in particular that of Bell 1990) and to concomitantly provide a transparent illustration and interpretation of the linear ingredients of the vortex's influence. In section 3 the nature of this influence is tested in the more realistic but still idealized atmospheric setting of nonlinear hemispheric primitive equation flow. This setup permits explicit examination of the jet–vortex interaction for vortices of different spatial scale, amplitude, and origin. Finally, in section 4 the derived results are placed in the context of extant studies of jet–vortex interaction and quasi-steady forcing of planetary- and subplanetary-scale atmospheric waves.

## 2. Interfacial waves on a *β* plane

### a. The flow setting

*β*plane with bottom topography

*η*=

*η*(

*x,*

*y*) and a rigid upper lid at a height

*H*(see Fig. 4) satisfies the following potential vorticity equation

*q*= {

*ξ*+ (

*f*

_{0}/

*H*)

*η*+

*βy*} is the potential vorticity,

*D*

_{g}/

*Dt*is a pseudo-Lagrangian derivative following the geostrophic flow,

*ξ*the vorticity, and

*ψ*denotes the corresponding geostrophic streamfunction (so that

*ξ*= ∇

^{2}

*ψ*). Pedlosky (1986) provides a resume of this flow setting.

*y*= 0 separating two semi-infinite domains of uniform but different values of quasigeostrophic potential vorticity,

*q*

*P*/2. The corresponding zonal flow field is given by

*q*

*υ*′ denotes the perturbation meridional velocity field and

*q*′ is the perturbation potential vorticity given by

### b. Free waves

*ω,*

*k*) denoting, respectively, the frequency and the zonal wavenumber. The dispersion relationship is obtained by integrating Eq. (3) across the PV interface, and is given by (see also Bell 1990; Swanson et al. 1997)

*k.*Consistent with the underlying Rossby wave character (i.e., propagation on a PV gradient) the individual modes propagate westward relative to the basic-state velocity field

*U*

_{0}at the interface. However, in contrast to Rossby waves on a uniform flow, the zonal group velocity of these waves is given by

*U*

_{0}and is independent of

*k.*Also, note that the dispersion relationship depends exclusively upon the in situ velocity and the vorticity jump at the interface, and not upon the other details of the background basic state.

*k*

_{s}such that

*U*

_{0}∼ 30 m s

^{−1}and mean basic-state vorticity values (

*P*/2) in the broad range (3–6) × 10

^{−5}s

^{−1}, the value of

*k*

_{s}equates in spherical geometry to a zonal wavenumber

*m*in the band of

*m*∼ 4–8. [A comparable relationship to Eq. (6) is derived in section 3a for the spherical geometry configuration.]

*k*satisfying Eq. (6). Alternatively for a forcing field advecting steadily eastward with a velocity

*U** the corresponding wavelength is given by

### c. Isolated topographic vortex forcing

*ψ*′ satisfies the relationship [see Eq. (4)]

*ψ*

_{a}and

*ψ*

_{b}), representing respectively a (possibly) time-dependent contribution due to perturbations of the interface (the

*q*′ term) and time-independent topographically induced component (the

*η*term).

*a,*ε) denoting the time-dependent amplitude and phase of the

*k*th wavenumber.

*η*) term consider an axisymmetric mountain of radius

*R*located at a distance

*d*(

*d*>

*R*) from the PV interface (see Fig. 4). The mountain-induced circumferential velocity

*r*) at a radius

*r*takes the form

*K*is a measure of the orographically induced vortex strength such that

*η*

_{0}. The associated perturbation meridional velocity at the interface can be written successively (using of the Fourier sine transform) as

*k*th wavenumber to the meridional velocity at the interface can be written in the form

*ψ*′, derived from using Eqs. (8) and (10), into Eq. (3) followed by integration across the interface at

*y*= 0 yields the following coupled set of equations for the time evolution of the

*k*th wave component (cf. Davies and Bishop 1994):

_{k}= −

*π*/2). The second equation (11) indicates that the net phase change arises from the difference between the westward propagation of a free mode (the

*γ*term) and the vortex-induced change. We note in passing that the nonlinear feedback due to the interfacial waves acting upon the topographically induced component is excluded in the linearized system represented by Eq. (3), but we return to consider this aspect in section 3.

It is instructive to consider two limit forms of the coupled set.

##### (i) A time-independent response:

##### (ii) The resonant response:

_{k}

*π*

*γ*

*a*

_{k}

*t*

*t.*

_{k}= −

*π*/2), and it is sustained because the corresponding free mode is stationary relative to the forcing (

*γ*= 0, cosε

_{k}= 0).

*a*

_{k}(0) = 0, is given by

_{k}

*μa*

_{k}

*ψ*

_{a}component of the streamfunction can be rewritten as

*k*).

Illustrations of the flow evolution for an elliptic-shaped mountain centered 1000 km poleward of the interface and with a major axis of 1000 km are provided in Fig. 5. (Note also that *H* = 10 km, *P* = 5 × 10^{−5} s^{−1}, *h*_{0} = 2 km, and a *β* plane with an east–west span of 27 000 km.) Figures 5a–c show the temporal development of the flow, and Fig. 5d the spatial pattern after 5 days. These depictions show a train of interfacial waves that evolve downstream with the ambient flow (sic. the group velocity), and their wavelength (∼8400 km) matches that of the stationary wave [Eq. (6)]. The wave train is well developed after 5 days, and after some 10 days the wave train approaches the “longitude” of the topography from the west.

*β*plane's east–west span equates to an integer multiple of the stationary wavelength. This behavior is illustrated (see Fig. 5c) by examining the time trace of normalized wave energy on the interface,

*E*(

*t*)/

*E*(0), where

*P,*

*U*

_{0}) corresponding to a potentially resonant (

*γ*= 0) setting and a nonresonant setting. Both experiments show a similar short-time evolution, but the longer-time development is markedly different. The energy for the “resonant” mode grows like

*t*

^{2}in the longer-time limit, whereas the amplitude of the off-resonant mode merely oscillates with time.

## 3. Interfacial waves on a hemisphere

The analysis of the previous section does not account for effects beyond *β*-plane geometry, quasigeostrophy, linearity, and a discontinuous PV interface. In this section we consider first the form of free waves on a barotropic PV discontinuity in spherical geometry and then examine the robustness of the dynamics using a nonlinear primitive equation model with a finite-width PV transition zone.

### a. Free waves

*P*poleward and zero equatorward of a specified latitude

*ϕ*

_{0}. The corresponding zonal velocity

*U*expressed in terms of

*U*cos

*ϕ*is then given by

*a*and Ω denoting the radius and angular velocity of the earth. Assuming a zero-perturbation meridional velocity at the equator, then the streamfunction in the two domains is given by

*λ,*

*m*) refer, respectively, to longitude and the zonal wavenumber,

*p*= tanh

^{−1}(sin

*ϕ*), and the amplitude modifying factor Γ is such that Γ = −

*e*

^{−mp0}

*mp*

_{0}).

_{0}is the angular velocity of the zonal flow at the latitude of the PV interface.

*β*-plane dispersion relationship is brought out on noting that for an interface located in midlatitudes

*mp*

_{0}

*m*≥ 3. In this limit the waves are in effect latitudinally trapped to the interface, the dispersion relationship reduces to

*ω*/∂

*m*is approximated by Ω

_{0}; that is, it is determined by the angular velocity at the interface.

*m*≥ 3) of the resonant waves prescribed by Eqs. (6) and (7) are given by

^{*}

_{0}

*P*and either Ω

_{0}or (Ω

_{0}−

^{*}

_{0}

*m.*In this context note that

*P*and Ω

_{0}are not interdependent since

*P*

_{0}

_{0}

*ϕ*

_{0}

*P*

_{0}

^{−1}

*ϕ*

_{0}

^{−1}

*P*

### b. Primitive equation simulations

Idealized numerical simulations are performed in a primitive equation (PE) framework to examine jet–vortex interaction on the hemisphere. The basic-state setting resembles that used in section 2.

#### 1) Model setup

Simulations are undertaken with an adiabatic version of the so-called Europa Model (EM) of the German Weather Service (Majewski 1991). This gridpoint model is used herein in a pole-centered, rotated-grid configuration on a quasi-hemispheric domain and operated with a 1° (∼100 km) resolution, 27 vertical levels, a free-slip lower boundary condition, a rigid upper-lid, weak fourth-order horizontal diffusion, and a relaxation zone in the equatorial region.

#### 2) Basic state

One component of the basic state in the extratropics is a barotropic zonal flow associated with two regions of uniform PV separated by a zonally aligned transition zone that extends over some 4° of latitude in the form of a column of strong horizontal PV gradient. The structure of the resulting barotropic jet and the accompanying vorticity distribution are shown in Fig. 6. This initial configuration bears comparison with the atmospheric settings discussed earlier that were portrayed in Figs. 1–4.

The other component comprises a vortexlike structure that takes one of the following three observationally motivated forms: (i) a single interior positive vorticity anomaly of meso-*α* scale (radius *a*_{0} ∼ 300 km) embedded within the high PV region (Fig. 6b) that is free to advect with the ambient flow and located at tropopause elevations at various distances from the transition zone, (ii) a counterpart positive or negative anomaly of synoptic-scale (*a*_{0} ∼ 700 km) (Fig. 9), and finally (iii) a hybrid setup with both an orographically bound negative anomaly and an advecting surface-based positive thermal anomaly.

These three configurations relate to different physical settings that were alluded to in the introduction (see also Fig. 3). Consider first the setting in case I with a *positive vortex* located poleward of and adjacent to a jet stream (cf. features marked A, B, and C in Fig. 3a). Both these ingredients have been implicated in the surface cyclogenesis. Incipient wavelike meanders of the jet stream (sic. ∇_{θ}PV waveguide) are often taken to be indicative of the upper-level signature of a troposphere-spanning growing baroclinic wave system (sic. baroclinic instability), and likewise a tropopause-level positive PV anomaly is noted as a possible precursor of surface frontal wave development. In effect these two categories bear comparison, respectively, with so-called Types A and B events of cyclogenesis.

In the present setup, consideration of interlevel interaction is eschewed and the focus was on single-level interaction. This is motivated by the recognition that frequent close proximity of a tropopause-level positive PV anomaly with the jet stream suggests that their interaction can indeed be a significant if not a dominant feature of the short-term development (see, e.g., Fehlmann and Davies 1999).

The setting in case II with a *negative synoptic-scale vortex* anomaly located within the high PV domain corresponds to the not-infrequent atmospheric setting when a large-scale upper-tropospheric air mass associated with a strong high pressure system has been sequestered into the stratosphere (cf. features marked D and E in Fig. 3a). The reverse can also occur with sequestration of stratospheric air into the upper troposphere in the form of an elongated PV streamer that can break up into a compact vortexlike structure (Appenzeller and Davies 1992). This is illustrated by the feature marked F in Fig. 3a.

Included in case III is the ingredient of perturbations on the ∇_{θ}PV band induced by a quasi-steady anticyclonic vorticity above topography (cf. the negative vorticity over the Rockies and Greenland in Fig. 3b). Other possible sources of such a steady forcing field would be regions of anomalous diabatic heating associated with SST anomalies in the subtropics. Again for a strong response the amplitude and scale of the forcing need to be appreciable and sustained, and concomitantly the ∇_{θ}PV band needs to be coherent. More trenchantly, on the longer time scale the possibility of a resonant response cannot be excluded a priori.

#### 3) Integral diagnostic

A description and discussion is now provided of the simulated patterns.

#### 4) Case I: Mesoscale vortex forcing

An interior, meso-*α* scale, positive PV anomaly is located some 14° poleward of a PV interface at 47.5°N. For this setting the component of the meridional velocity within the transition zone attributable to the vortex forcing is comparatively weak and the response should be at most quasilinear.

Figure 7 shows the perturbation vorticity field on the 500-hPa surface for the period 60 to 300 hours. In accord with linear theory a trapped wave train of PV anomalies develops within the transition zone of high PV gradient. It evolves downstream with the background in situ flow speed, and its wavenumber (*m* = 6) conforms to that expected from Eq. (20).

Consider the amplitude and structure of the induced patterns. First, note that the wave train is confined to the near neighborhood of the PV transition zone. In effect, nonlinear effects are small and the perturbation energy is exported downstream rather than being available ab initio to build up the in situ perturbation amplitude. Indeed the wave train's vorticity anomalies are ∼0.25 × 10^{−4} s^{−1} and this amounts to only a third of the vorticity difference across the transition zone.

Second, note that the backreaction (Bell 1990) of the wave train upon the vortex's latitudinal movement appears to be weak. Indeed the vortex advects eastward at a quasi-constant latitude in the time period beyond 60 hours. From a synoptic-dynamic standpoint it is evident that the prevailing configuration is such that the isolated vortex is located only slightly to the west of the first (negative) vorticity perturbation on the wave train, and concomitantly the meridional velocity at the vortex center is comparatively weak (not shown). In effect, the realized phase is not supportive of a strong backreaction of the wave train upon the vortex's movement. From a global-integrative standpoint further insight on the weak backreaction can be gleaned from Eq. (21). The wave train's weak amplitude and its confinement to the vicinity of the transition zone implies that the lateral movement of the vortex required by Eq. (21) does not need to be appreciable.

Third, the isolated vortex retains a striking coherence throughout the simulation despite the latitudinal shear of the ambient flow. This is another demonstration that nonlinear dynamical effects can enable a vortex to counter the deforming effects of an ambient shear (Meacham et al. 1990).

Fourth, the wave train encircles the globe after 285 h and this is followed by a measure of constructive interference with the *m* = 6 pattern persisting at least until 480 h (not shown). This longer-term behavior is the manifestation of the resonant buildup of the wave train's amplitude and accords with the linear theory of the previous section. However, it also points to the theory's limited validity and applicability. A temporal buildup of the wave amplitude also requires [see Eq. (21)] a corresponding movement of the vortex away from the critical latitude [defined by Eq. (20)], and this implies that the longer-term response can at most be a nonsingular resonance.

The sensitivity of the nature of the response to the vortex's latitude [cf. Eq. (20)] is brought out by conducting three simulations with the vortex placed at different initial latitudes corresponding approximately to constructive interference at zonal wavenumbers of *m* = 5 and 6, and an intermediate destructive setting. Figure 8 shows the perturbation vorticity pattern on the 500-hPa surface for each simulation at a time instance (*t* = 240 h) shortly before the wave train has encircled the globe and at a later instant (*t* = 315 h). It is evident that destructive interference prevails downstream in the off-resonant setting.

#### 5) Case II: Synoptic-scale vortex forcing

Counterpart simulations to case I are performed with first a positive and then a negative synoptic-scale vortex (*a*_{0} ∼ 700 km). The scale and strength of the anomalies betoken significantly larger initial velocity field at the interface that is attributable to the vortex, and thereby the likelihood of nonlinear effects influencing the subsequent flow evolution. Figures 9a and 9b show, for the positive and negative anomalies respectively, the perturbation PV fields at tropopause elevations for the initial time and at 24, 36, 48, and 72 h.

The flow evolution differs markedly for the two cases, but there are three common underlying dynamical effects. First, both vortices undergo a strong asymmetric deformation under the influence of the ambient flow's lateral shear while perturbations evolve on the interface. During this phase the more equatorward portion of the vortex becomes distended longitudinally to form an elongated filament.

Second, the distorted vortex's own velocity signal induces in the case of the positive (negative) anomaly a poleward (equatorward) displacement of the filament away from (toward) the PV interface and thereby serves to weaken (enhance) the filament's interaction with anomalies evolving on the interface. The sign of the displacement is in accord with Eq. (21).

Third, there is a tendency for the vortex to become aligned with an adjacent oppositely signed anomaly on the interface whereas, in contrast, the filament becomes juxtaposed to or coalesces with a similarly signed interface anomaly.

In the subsequent evolution (not shown) the positive vortex sheds its filament, regains its circular shape at a more poleward latitude, and thereafter a wave train develops on the interface as in case I. In contrast the negative anomaly approaches the interface and eventually is smeared out. Thus the finite amplitude developments in the two cases are radically different but consistent with the integral invariant [Eq. (21)].

#### 6) Case III: Hybrid vortex forcing

In this third case Greenland-scale and height topography is located poleward of the interface. The initial conditions are such that the perturbing effect of the topography induces two vortexlike structures that subsequently proceed to influence the PV transition zone. One vortex is the topographically bound anticyclone (cf. the forcing discussed for the *β*-plane setting of section 2). In effect this vortex results from the fission of the incident potential vorticity into a negative relative vorticity compensated by enhanced thermal stratification as the flow surmounts the terrain (cf. Schwierz and Davies 2003).

The second vortex is shed off the topography in the first phase of the simulation and advects with the ambient flow. It is a surface-based positive thermal anomaly resulting from the imposed initial conditions of uniform stratification and possesses a concomitant positive vorticity.

In this case (Fig. 10) a composite pattern develops with a short-wavelength wave train (*m* ∼ 6) evolving downstream of the advecting vortex and a longer-wavelength and weaker-amplitude wave train (*m* ∼ 3) evolving downstream from the stationary topographically bound anticyclone. The difference in wavelength is in accord with Eqs. (19) and (20) with the smaller relative velocity of the advecting vortex associated with a shorter-wavelength wave train. The difference in vortical amplitude accords with the reduction of the energy flux transferred to the stationary wave train.

An alternative depiction of the development is shown in Fig. 11 in the form of a Hovmöller diagram. The development occurs in three phases. An initial development of the longer-wavelength features followed by the codevelopment of the almost spatially separate wave trains, and finally the emergence of an interference pattern (with *m* ∼ 5) after the short-wavelength wave train approaches the longitude of the topography.

#### 7) Further remarks

In case I the simulations with the mesoscale vortex indicates that a major distortion of the jet conducive to deep cyclogenesis requires stronger vortex forcing. Such forcing with concomitant in situ development, rather than energy propagation downstream along the waveguide, could ensue [see Eq. (21)] with a pairing of the positive anomaly and a negative anomaly created on the interface and their subsequent poleward displacement (cf. case II).

In case II the simulations involved strong nonlinear effects and there was a tendency for the negative vortex to move toward, and conceivably for a larger amplitude vortex to be extruded into the low PV domain. Such an expulsion is often observed on isentropic PV charts and the result has a bearing upon the assessment of net stratosphere–troposphere exchange of mass and chemical constituents.

For case III the simulation showed that the ∇_{θ}PV band can indeed serve as a planetary-scale waveguide transmitting the effects of the forcing to the far field along a well-defined and confined path. Note on the other hand (cf. case II) that a large amplitude forcing can deform the band irreversibly and thereby hinder propagation downstream.

## 4. Final remarks

The present study was predicated upon the assertion that the distinctive structure of the PV pattern in the neighborhood of the tropopause-level jet stream can exert a major influence upon large-scale atmospheric flow. In particular, it was conjectured that the accompanying band of highly enhanced ∇_{θ}PV can serve as an effective waveguide for large-scale atmospheric flow and the seat for trapped Rossby waves forced by juxtaposed isolated PV anomalies and/or larger-scale orography.

These conjectures were tested in an idealized setting comprising a vortexlike forcing of a zonally aligned PV discontinuity. First, the quintessential dynamics of the vortex's influence upon the PV interface were elucidated in the linear barotropic *β*-plane limit. In this setting trapped Rossby waves can be sustained on the interface, and our analysis pointed to the factors required to produce a large response to the forcing. These included planetary- or subplanetary-scale steady-state forcing or synoptic-scale forcing for an advecting vortex, the possible establishment of a resonant response in the longer-time frame for a suitable large-scale flow setting and a vortex moving zonally at the Doppler-shifted velocity of a trapped Rossby wave.

Thereafter further aspects of the interaction were examined in a hemispheric primitive equation setting using a nonlinear numerical model. Simulations performed with various forcing configurations served to lend credence to nonsingular resonant interaction in a setting with weak forcing and to indicate the disparate range of development patterns in a setting with stronger forcing.

Clearly the generality and applicability of the derived results is limited by the selection of a simple idealized setting. In reality, the domain of enhanced PV gradient on isentropic surfaces is a narrow tube that meanders around the globe and is often breached by large-amplitude breaking waves rather than taking the form of a zonally and vertically aligned interface (see Figs. 1 and 3). Moreover, the presence of both the tropopause-level tube of enhance PV gradient plus the narrow zone of enhanced surface baroclinicity (excluded herein) constitute two waveguides and, as in classical baroclinic instability, perturbations on these waveguides can interact to their mutual enhancement.

Nevertheless the derived results do bear comparison with and shed light on observed phenomena. To set the results in context it is appropriate to note that the present study's spatially confined Rossby waves are trapped internally by the atmosphere's dynamical (PV) structure and therefore differ intrinsically from classical Rossby waves propagating on comparatively smooth PV fields. Moreover, the markedly weak isentropic PV gradients away from the jet limits the efficacy of meridional wave propagation and the concomitant energy flux of conventional Rossby waves.

For synoptic-scale flow one inference is that the initiation of major upper-level wave development would be favored by a large amplitude, suitably scaled, and latitudinally located PV anomaly. The latitude corresponds to a location where the vortex would advect eastward with a zonal velocity matching that of the trapped wave. It is pertinent to record that such anomalies are more likely to be located at the tropopause level. Again, subplanetary-scale (*m* ∼ 3–7) steady-state forcing could be effective for a suitable large-scale flow setting. In this case an informative and illustrative depiction of the markedly different nature of wave propagation in superrotational flow versus one with a localized jet structure is shown in Branstator (1985a, Fig. 3). This result points to the efficacy of the ∇_{θ}PV band's role as a waveguide for subplanetary–scale waves. The propagation on a PV waveguide is also in accord with simulations performed with a barotropic zonally varying basic state derived from the climatological Northern Hemisphere 300-hPa field (Blackmon et al. 1983; Branstator 1985b, 2002) and with the more regularly structured waves observed on the Southern Hemisphere jet (Simmons et al. 1983). In a related vein it is pertinent to note that the subplanetary scale carries most of the signal of the atmosphere's interannual variability, and Massacand and Davies (2001) interpreted the observed patterns of this variability directly in terms of wave propagation along the time-mean ∇_{θ}PV band—in effect the PV waveguide. Hence, the analysis outlined here lends further support to a complementary interpretation of the establishment of teleconnection patterns.

## Acknowledgments

The authors would like to express their gratitude to René Fehlmann for providing the PV inversion routine and support with an early version of the simulation setup and to Daniel Lüthi for expert help with the EM model. The neat deduction leading to Eq. (16) was pointed out to us by one of the reviewers. We thank ECMWF and MeteoSwiss for providing access to the meteorological data. This study was conducted in part with funding from the NCCR Climate program of the Swiss National Science Foundation.

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Mean winter (DJF) isentropic PV gradient for the 15-yr period 1979–93 (ERA-15 data). (a) On 320 K with wind speed (contours 30, 40, 50 m s^{−1}). (b) Cross section at 75°W; 30 and 40 m s^{−1} isotachs (dashed) overlaid. Same as Fig. 1, but note the difference in scale and magnitude

Citation: Journal of the Atmospheric Sciences 61, 1; 10.1175/1520-0469(2004)061<0073:FWOAZA>2.0.CO;2

Mean winter (DJF) isentropic PV gradient for the 15-yr period 1979–93 (ERA-15 data). (a) On 320 K with wind speed (contours 30, 40, 50 m s^{−1}). (b) Cross section at 75°W; 30 and 40 m s^{−1} isotachs (dashed) overlaid. Same as Fig. 1, but note the difference in scale and magnitude

Citation: Journal of the Atmospheric Sciences 61, 1; 10.1175/1520-0469(2004)061<0073:FWOAZA>2.0.CO;2

Mean winter (DJF) isentropic PV gradient for the 15-yr period 1979–93 (ERA-15 data). (a) On 320 K with wind speed (contours 30, 40, 50 m s^{−1}). (b) Cross section at 75°W; 30 and 40 m s^{−1} isotachs (dashed) overlaid. Same as Fig. 1, but note the difference in scale and magnitude

Citation: Journal of the Atmospheric Sciences 61, 1; 10.1175/1520-0469(2004)061<0073:FWOAZA>2.0.CO;2

(a) Instantaneous PV on the 320-K isentrope at 1800 UTC 24 Dec 1999 (shaded, in PVU) with 2-PVU PV contour (bold black) and synoptic and mesoscale vortex anomalies (labels A–F) overlaid; (b) 15-yr winter (DJF) mean of negative vorticity on 500 hPa (shaded, units 10^{−4} s^{−1})

Citation: Journal of the Atmospheric Sciences 61, 1; 10.1175/1520-0469(2004)061<0073:FWOAZA>2.0.CO;2

(a) Instantaneous PV on the 320-K isentrope at 1800 UTC 24 Dec 1999 (shaded, in PVU) with 2-PVU PV contour (bold black) and synoptic and mesoscale vortex anomalies (labels A–F) overlaid; (b) 15-yr winter (DJF) mean of negative vorticity on 500 hPa (shaded, units 10^{−4} s^{−1})

Citation: Journal of the Atmospheric Sciences 61, 1; 10.1175/1520-0469(2004)061<0073:FWOAZA>2.0.CO;2

(a) Instantaneous PV on the 320-K isentrope at 1800 UTC 24 Dec 1999 (shaded, in PVU) with 2-PVU PV contour (bold black) and synoptic and mesoscale vortex anomalies (labels A–F) overlaid; (b) 15-yr winter (DJF) mean of negative vorticity on 500 hPa (shaded, units 10^{−4} s^{−1})

Citation: Journal of the Atmospheric Sciences 61, 1; 10.1175/1520-0469(2004)061<0073:FWOAZA>2.0.CO;2

Schematic diagram to illustrate the analytic model setup, terms, and symbols. (top) A vertical section at *y* = *d.* See text for details.

Citation: Journal of the Atmospheric Sciences 61, 1; 10.1175/1520-0469(2004)061<0073:FWOAZA>2.0.CO;2

Schematic diagram to illustrate the analytic model setup, terms, and symbols. (top) A vertical section at *y* = *d.* See text for details.

Citation: Journal of the Atmospheric Sciences 61, 1; 10.1175/1520-0469(2004)061<0073:FWOAZA>2.0.CO;2

Schematic diagram to illustrate the analytic model setup, terms, and symbols. (top) A vertical section at *y* = *d.* See text for details.

Citation: Journal of the Atmospheric Sciences 61, 1; 10.1175/1520-0469(2004)061<0073:FWOAZA>2.0.CO;2

(a) Initial velocity *υ*^{′}_{a}*x* axis are 1000 km. (b) Amplitude evolution for different modes *k* in the domain of length *L*_{ϕ} = 2*πR* cos*ϕ,* with *R* (earth's radius) and Δ*k* = 2*π*/*L*_{ϕ}: 3Δ*k* (dotted), Δ*k* (bold) and 5Δ*k* (light). Time in days. (c) Energy evolution [normalized, *E*(*t*)/*E*(0)] for the two extreme cases *k*_{r} = 3Δ*k* (bold) and *k*_{r} = 3.5Δ*k* (light). Also shown (dotted) is a function proportional to *t*^{2} (for *t* ≥ 10 days). (d) Spatial flow pattern after 5 days

Citation: Journal of the Atmospheric Sciences 61, 1; 10.1175/1520-0469(2004)061<0073:FWOAZA>2.0.CO;2

(a) Initial velocity *υ*^{′}_{a}*x* axis are 1000 km. (b) Amplitude evolution for different modes *k* in the domain of length *L*_{ϕ} = 2*πR* cos*ϕ,* with *R* (earth's radius) and Δ*k* = 2*π*/*L*_{ϕ}: 3Δ*k* (dotted), Δ*k* (bold) and 5Δ*k* (light). Time in days. (c) Energy evolution [normalized, *E*(*t*)/*E*(0)] for the two extreme cases *k*_{r} = 3Δ*k* (bold) and *k*_{r} = 3.5Δ*k* (light). Also shown (dotted) is a function proportional to *t*^{2} (for *t* ≥ 10 days). (d) Spatial flow pattern after 5 days

Citation: Journal of the Atmospheric Sciences 61, 1; 10.1175/1520-0469(2004)061<0073:FWOAZA>2.0.CO;2

(a) Initial velocity *υ*^{′}_{a}*x* axis are 1000 km. (b) Amplitude evolution for different modes *k* in the domain of length *L*_{ϕ} = 2*πR* cos*ϕ,* with *R* (earth's radius) and Δ*k* = 2*π*/*L*_{ϕ}: 3Δ*k* (dotted), Δ*k* (bold) and 5Δ*k* (light). Time in days. (c) Energy evolution [normalized, *E*(*t*)/*E*(0)] for the two extreme cases *k*_{r} = 3Δ*k* (bold) and *k*_{r} = 3.5Δ*k* (light). Also shown (dotted) is a function proportional to *t*^{2} (for *t* ≥ 10 days). (d) Spatial flow pattern after 5 days

Citation: Journal of the Atmospheric Sciences 61, 1; 10.1175/1520-0469(2004)061<0073:FWOAZA>2.0.CO;2

(a) Zonal velocity profile (solid line, units m s^{−1}) and vorticity discontinuity [dashed, values 7.25 and 15 (×10^{−5} s^{−1})] of the barotropic initial state for the PE model operated on the sphere. The PV discontinuity is located at 47.5°N. (b) Initial positive vorticity anomaly (shaded, units 10^{−4} s^{−1}) located north of the PV discontinuity (solid line).

Citation: Journal of the Atmospheric Sciences 61, 1; 10.1175/1520-0469(2004)061<0073:FWOAZA>2.0.CO;2

(a) Zonal velocity profile (solid line, units m s^{−1}) and vorticity discontinuity [dashed, values 7.25 and 15 (×10^{−5} s^{−1})] of the barotropic initial state for the PE model operated on the sphere. The PV discontinuity is located at 47.5°N. (b) Initial positive vorticity anomaly (shaded, units 10^{−4} s^{−1}) located north of the PV discontinuity (solid line).

Citation: Journal of the Atmospheric Sciences 61, 1; 10.1175/1520-0469(2004)061<0073:FWOAZA>2.0.CO;2

(a) Zonal velocity profile (solid line, units m s^{−1}) and vorticity discontinuity [dashed, values 7.25 and 15 (×10^{−5} s^{−1})] of the barotropic initial state for the PE model operated on the sphere. The PV discontinuity is located at 47.5°N. (b) Initial positive vorticity anomaly (shaded, units 10^{−4} s^{−1}) located north of the PV discontinuity (solid line).

Citation: Journal of the Atmospheric Sciences 61, 1; 10.1175/1520-0469(2004)061<0073:FWOAZA>2.0.CO;2

Time evolution (60–300 h) of the perturbation vorticity response (shaded, units 10^{−4} s^{−1}) for a moving vorticity anomaly acting on a vorticity gradient at 47.5°N. Latitude circles shown for 40° and 80°N.

Citation: Journal of the Atmospheric Sciences 61, 1; 10.1175/1520-0469(2004)061<0073:FWOAZA>2.0.CO;2

Time evolution (60–300 h) of the perturbation vorticity response (shaded, units 10^{−4} s^{−1}) for a moving vorticity anomaly acting on a vorticity gradient at 47.5°N. Latitude circles shown for 40° and 80°N.

Citation: Journal of the Atmospheric Sciences 61, 1; 10.1175/1520-0469(2004)061<0073:FWOAZA>2.0.CO;2

Time evolution (60–300 h) of the perturbation vorticity response (shaded, units 10^{−4} s^{−1}) for a moving vorticity anomaly acting on a vorticity gradient at 47.5°N. Latitude circles shown for 40° and 80°N.

Citation: Journal of the Atmospheric Sciences 61, 1; 10.1175/1520-0469(2004)061<0073:FWOAZA>2.0.CO;2

Perturbation vorticity response at wavenumbers (top) *m* = 5, (bottom) *m* = 6, and (middle) for a nonresonant case for the integration times (left) 240 h and (right) 315 h. The different wavenumbers result from the particular location of the anomaly (bold contours 0.1, 0.3, 0.5 PVU) relative to the waveguide (65.5°, 63°, and 61.5°N, respectively). Cylindrical projection of the region 40°–75°N, 180°W–180°E.

Citation: Journal of the Atmospheric Sciences 61, 1; 10.1175/1520-0469(2004)061<0073:FWOAZA>2.0.CO;2

Perturbation vorticity response at wavenumbers (top) *m* = 5, (bottom) *m* = 6, and (middle) for a nonresonant case for the integration times (left) 240 h and (right) 315 h. The different wavenumbers result from the particular location of the anomaly (bold contours 0.1, 0.3, 0.5 PVU) relative to the waveguide (65.5°, 63°, and 61.5°N, respectively). Cylindrical projection of the region 40°–75°N, 180°W–180°E.

Citation: Journal of the Atmospheric Sciences 61, 1; 10.1175/1520-0469(2004)061<0073:FWOAZA>2.0.CO;2

Perturbation vorticity response at wavenumbers (top) *m* = 5, (bottom) *m* = 6, and (middle) for a nonresonant case for the integration times (left) 240 h and (right) 315 h. The different wavenumbers result from the particular location of the anomaly (bold contours 0.1, 0.3, 0.5 PVU) relative to the waveguide (65.5°, 63°, and 61.5°N, respectively). Cylindrical projection of the region 40°–75°N, 180°W–180°E.

Citation: Journal of the Atmospheric Sciences 61, 1; 10.1175/1520-0469(2004)061<0073:FWOAZA>2.0.CO;2

(a) Positive and (b) negative synoptic-scale vortex anomalies (*a*_{0} ∼ 700 km, lat_{0} ∼ 61°N) acting on a discontinuity at 47.5°N. Perturbation isentropic PV on 320 K for 0, 24, 36, 48, and 72 h. Contour interval is 0.15 PVU. Vortex anomaly amplitudes at initial time are 12 and −1.5 PVU, respectively

Citation: Journal of the Atmospheric Sciences 61, 1; 10.1175/1520-0469(2004)061<0073:FWOAZA>2.0.CO;2

(a) Positive and (b) negative synoptic-scale vortex anomalies (*a*_{0} ∼ 700 km, lat_{0} ∼ 61°N) acting on a discontinuity at 47.5°N. Perturbation isentropic PV on 320 K for 0, 24, 36, 48, and 72 h. Contour interval is 0.15 PVU. Vortex anomaly amplitudes at initial time are 12 and −1.5 PVU, respectively

Citation: Journal of the Atmospheric Sciences 61, 1; 10.1175/1520-0469(2004)061<0073:FWOAZA>2.0.CO;2

(a) Positive and (b) negative synoptic-scale vortex anomalies (*a*_{0} ∼ 700 km, lat_{0} ∼ 61°N) acting on a discontinuity at 47.5°N. Perturbation isentropic PV on 320 K for 0, 24, 36, 48, and 72 h. Contour interval is 0.15 PVU. Vortex anomaly amplitudes at initial time are 12 and −1.5 PVU, respectively

Citation: Journal of the Atmospheric Sciences 61, 1; 10.1175/1520-0469(2004)061<0073:FWOAZA>2.0.CO;2

Time evolution (60–300 h) of the perturbation vorticity response (shaded, units 10^{−4} s^{−1}) for both an orographic stationary vorticity source (dashed contours) and a moving vorticity anomaly (solid contours) acting on a vorticity gradient at 47.5°N.

Citation: Journal of the Atmospheric Sciences 61, 1; 10.1175/1520-0469(2004)061<0073:FWOAZA>2.0.CO;2

Time evolution (60–300 h) of the perturbation vorticity response (shaded, units 10^{−4} s^{−1}) for both an orographic stationary vorticity source (dashed contours) and a moving vorticity anomaly (solid contours) acting on a vorticity gradient at 47.5°N.

Citation: Journal of the Atmospheric Sciences 61, 1; 10.1175/1520-0469(2004)061<0073:FWOAZA>2.0.CO;2

Time evolution (60–300 h) of the perturbation vorticity response (shaded, units 10^{−4} s^{−1}) for both an orographic stationary vorticity source (dashed contours) and a moving vorticity anomaly (solid contours) acting on a vorticity gradient at 47.5°N.

Citation: Journal of the Atmospheric Sciences 61, 1; 10.1175/1520-0469(2004)061<0073:FWOAZA>2.0.CO;2

Vorticity Hovmöller diagram at 47.5°N for a 480-h PE simulation (PV discontinuity located at 47.5°N, orography centered at 45°W)

Citation: Journal of the Atmospheric Sciences 61, 1; 10.1175/1520-0469(2004)061<0073:FWOAZA>2.0.CO;2

Vorticity Hovmöller diagram at 47.5°N for a 480-h PE simulation (PV discontinuity located at 47.5°N, orography centered at 45°W)

Citation: Journal of the Atmospheric Sciences 61, 1; 10.1175/1520-0469(2004)061<0073:FWOAZA>2.0.CO;2

Vorticity Hovmöller diagram at 47.5°N for a 480-h PE simulation (PV discontinuity located at 47.5°N, orography centered at 45°W)

Citation: Journal of the Atmospheric Sciences 61, 1; 10.1175/1520-0469(2004)061<0073:FWOAZA>2.0.CO;2